Calculating the Cosmos-How Mathematics Unveils the Universe

1979. We now know that Jupiter and Neptune also have very faint ring systems. Saturn’s moon Rhea may have its own very tenuous ring system. Moreover, Douglas Hamilton and Michael Skrutskie discovered in 2009 that Saturn has an absolutely gigantic, but very faint, ring, far larger than the ones Galileo and the Voyagers saw. They missed it, in part, because it’s visible only in infrared light. Its inner edge is about 6 million kilometres from the planet, and the outer edge is 18 million kilometres distant. The moon Phoebe orbits inside it, and may well be responsible for it. It’s very tenuous, made from ice and dust, and it may help to solve a long-standing mystery: Iapetus’s dark side. One half of the moon Iapetus is brighter than the other, an observation that has puzzled astronomers since about 1700 when Cassini first noticed it. The suggestion is that Iapetus is sweeping up dark material from the giant ring. In 2015 Matthew Kenworthy and Eric Mamajek announced2 that a distant exoplanet, orbiting the star J1407, has a system of rings that makes Saturn’s pale into insignificance, even taking the newest one into account. The discovery, like the rings of Uranus, is based on fluctuations observed in a light curve – the main way to locate exoplanets (see Chapter 13). As the planet crosses the star (transits) the star’s light dims. In this instance, it dimmed repeatedly, over a two-month period, but each dimming event was fairly rapid. The inference is that some exoplanet with numerous rings must have been passing across the path from the star to the Earth. The best ring model has 37 rings and extends out to a radius of 0·6 AU (90 million km). The exoplanet itself hasn’t yet been detected, but it’s thought to be about 10–40 times as massive as Jupiter. A clear gap in the ring system is most readily explained by the presence of an exomoon, whose size can also be estimated. In 2014 another ring system was discovered in the solar system in an improbable place: around (10199) Chariklo, a type of small body known as a centaur. 3 This one orbits between Saturn and Uranus, and it’s the largest known centaur. Its rings showed up as two slight dips of brightness in a series of observations in which Chariklo obscured (jargon: occulted) various stars. The relative positions of these dips are all close to the same ellipse, with Chariklo near the centre, suggesting two nearby rings in fairly circular orbits, whose plane is being viewed at an angle. One has radius 391

km and is about 7 km across; then there’s a 9 km gap to the second, at radius 405 km. Since ring systems occur repeatedly, they can’t just be an accident. So how do ring systems form? There are three main theories. They may have formed when the original gas disc coalesced to create the planet; they could be relics of a moon that has been broken by a collision; they could be remains of a moon that got closer than the Roche limit, at which tidal forces exceed the strength of the rock, and broke up. Catching a ring system during its formation is unlikely, although Kenworthy and Mamajek’s discovery shows that it’s possible, but the best that it can give us is a snapshot. Observing the entire process would take hundreds of lifetimes. What we can do, though, is to analyse hypothetical scenarios mathematically, make predictions, and compare them with observations. It’s like fossil-hunting in the heavens. Each ‘fossil’ provides evidence for what happened in the past, but you need a hypothesis to interpret the evidence, and you need mathematical simulations or inferences or, better still, theorems to understand the consequences of that hypothesis.

7 Cosimo’s Stars Since it is up to me, the first discoverer, to name these new planets, I wish, in imitation of the great sages who placed the most excellent heroes of that age among the stars, to inscribe these with the name of the Most Serene Grand Duke [Cosimo II de’ Medici, Grand Duke of Tuscany]. Galileo Galilei, Sidereus Nuncius WHEN GALILEO FIRST OBSERVED JUPITER through his new telescope, he noticed four tiny specks of light in orbit around the planet, so Jupiter had moons of its own. This was direct evidence that the geocentric theory must be wrong. He sketched their arrangements in his notebook. More detailed observations can be strung together, so we can plot the paths along which the specks seem to move. When we do, we obtain beautiful sine curves. The natural way to generate a sine curve is to observe uniform circular motion sideways on. So Galileo inferred that Cosimo’s stars go round Jupiter in circles, in the plane of the ecliptic. Improved telescopes revealed that most planets in the solar system have moons. Mercury and Venus are the sole exceptions. We’ve got 1, Mars has 2, Jupiter has at least 67, Saturn at least 62 plus hundreds of moonlets, Uranus 27, and Neptune 14. Pluto has 5. The satellites range from small irregular rocks to nearly spherical ellipsoids big enough to be small planets. Their surfaces can be mainly rock, ice, sulphur, or frozen methane. Mars’s tiny moons Phobos and Deimos race across the Martian sky, and Phobos is so close that it moves in the opposite direction to Deimos. Both bodies are irregular, and are probably captured asteroids – or, possibly, a captured asteroid with a duck shape like comet 67P, recently shown to be two bodies that came gently together and stuck. If so, the one that Mars captured came unstuck again because of the planet’s gravity; Phobos is one piece and Deimos the other.

Galileo’s records of the moons. Positions of Jupiter’s moons as seen from Earth, forming sine curves. Some moons seem totally dead; others are active. Saturn’s moon Enceladus produces towering ice geysers 500 kilometres high. Jupiter’s moon Io has a sulphurous surface and at least two active volcanos, Loki and Pele, spewing out sulphur compounds. There must be huge subsurface

reservoirs of liquid sulphur, and the energy to heat it probably comes from gravitational squeezing by Jupiter. Saturn’s moon Titan has a methane atmosphere, much denser than it ought to be. Neptune’s moon Triton is going round the planet the wrong way, indicating that it was captured. It’s spiralling slowly inwards, and 3·6 billion years from now it will break up when it hits the Roche limit, the distance inside which moons come to pieces under gravitational stress. The moons of the larger planets often exhibit resonances. For example, Europa has twice the period of Io, and Ganymede has twice the period of Europa, hence four times that of Io. Resonances come from the dynamics of bodies obeying Newton’s law of gravity. Planets with ring systems slowly accrete moons at the edge of the rings, and then ‘spit them out’ one by one, like water dripping from a tap. There are mathematical regularities in this process. Various lines of evidence, some mathematical, indicate that several icy moons have underground oceans, melted by tidal forces. At least one contains more water than all Earth’s oceans combined. The presence of liquid water makes them potential habitats for simple Earthlike life forms, see Chapter 13. And Titan’s unusual chemistry might make it a potential habitat for un-Earthlike life forms. At least one asteroid has a very tiny moon of its own: Ida, orbited by diminutive Dactyl. Moons are fascinating, a playground for gravitational modelling and scientific speculation of all kinds. And it all goes back to Galileo and Cosimo’s stars.

Asteroid Ida (left) and its moon Dactyl (right). In 1612, when Galileo had determined the orbital periods of Cosimo’s stars, he suggested that sufficiently accurate tables of their motion would provide a clock in the sky, solving the longitude problem in navigation. At that time navigators could estimate their latitude by observing the Sun (though accurate instruments like the sextant were far in the future), but longitude relied on dead reckoning – educated guesswork. The main practical issue was to make observations from the deck of a ship as it tossed on the waves, and he worked on two devices for stabilising a telescope. The method was used on land but not at sea. John Harrison famously solved the longitude problem with his series of very accurate chronometers, eventually being awarded prize money in 1773. Jupiter’s moons presented astronomers with a celestial laboratory, allowing them to observe systems of several bodies. They tabulated their movements and attempted to explain and predict them theoretically. One way to obtain precise measurements is to observe a transit of a moon across the face of the planet, because the start and end of a transit are well-defined events. Eclipses, where the moon goes behind the planet, are similarly well defined. Giovanni Hodierna said as much in 1656, and a decade or so later Cassini began a lengthy series of systematic observations, noting other coincidence such as conjunctions, in which two of the moons appear to be aligned. To his surprise, the transit times didn’t seem consistent with the moons moving in regular, repetitive orbits.

The Danish astronomer Ole Rømer took up Galileo’s suggestion about longitude, and in 1671 he and Jean Picard observed 140 eclipses of Io from Uraniborg, near Copenhagen, while Cassini did the same from Paris. Comparing the timings, they calculated the difference between the longitudes of those two locations. Cassini had already noticed some peculiarities in the observations, and wondered whether they resulted from light having a finite speed. Rømer combined all the observations, and discovered that the time between successive eclipses became shorter when Earth was closer to Jupiter, and longer when it was further away. In 1676 he informed the Academy of Sciences of the reason: ‘Light seems to take about ten to eleven minutes [to cross] a distance equal to the radius of the terrestrial orbit.’ This figure relied on some careful geometry, but the observations were inaccurate; the modern value is 8 minutes 12 seconds. Rømer never published his results as a formal paper, but the lecture was summarised – badly – by an unknown reporter. Scientists didn’t accept that light has a finite speed until 1727. Despite the irregularities, Cassini never observed a triple conjunction of the inner moons, Io, Europa, and Ganymede – all three of them aligning simultaneously – so something must prevent this. Their orbital periods are roughly in the ratio 1:2:4, and in 1743 Pehr Wargentin, Director of the Stockholm Observatory, showed that this relationship becomes astonishingly accurate if it’s reinterpreted correctly. Measuring their positions as angles relative to a fixed radius, he discovered a remarkable relation: angle for Io – 3 × angle for Europa + 2 × angle for Ganymede = 180° According to his observations, this equation holds almost exactly over long periods of time, despite the irregularities in the three moons’ orbits. A triple conjunction requires the three angles to be equal, but if they are, the left-hand side of this equation is 0°, not 180°. So a triple conjunction is impossible as long as the relationship holds good. Wargentin stated that it wouldn’t happen for at least 1·3 million years. The equation also implies a specific pattern to conjunctions of these moons, which occur in a repeating cycle:

Europa with Ganymede Io with Ganymede Io with Europa Io with Ganymede Io with Europa Io with Ganymede Successive conjunctions of the three innermost moons of Jupiter: Io, Europa, and Ganymede (reading outwards). Laplace decided that Wargentin’s formula couldn’t be coincidence, so there must be a dynamical reason. In 1784 he deduced the formula from Newton’s law of gravitation. His calculations imply that over long periods the combination of angles concerned does not remain exactly at 180°; instead, it librates – oscillates slowly either side of that value – by less than 1°. This is small enough to prevent a triple conjunction. He predicted the period of this oscillation to be 2270 days. The observed figure today is 2071 days, not bad. In his honour the relationship between the three angles is called the Laplace resonance. His success was a significant confirmation of Newton’s law. We now know why the transit times are irregular. Jupiter’s gravity causes the approximately elliptical orbits of its moons to precess (like Mercury’s orbit round the Sun), so that the position of perijove – the closest approach to Jupiter – changes quite fast. In the Laplace resonance formula these precessions cancel out, but they have a strong effect on individual transits.

Any similar relationship is also called a Laplace resonance. The star Gliese 876 has a system of exoplanets, the first being found in 1998. Four are now known, and three of them – Gliese 876c, Gliese 876b, and Gliese 876e – have orbital periods of 30·008, 61·116, and 124·26 days, suspiciously close to ratios 1:2:4. In 2010 Eugenio Rivera and colleagues1 showed that in this case the relationship is: angle for 876c − 3 × angle for 876b + 2 × angle for 876e = 0° but the sum librates around 0° by as much as 40°, a far bigger oscillation. Now triple conjunctions are possible, and near triple conjunctions occur once for every revolution of the outermost planet. Simulations indicate that the oscillation around 0° should be chaotic, with an approximate period of about ten years. Three of Pluto’s moons – Nix, Styx, and Hydra – exhibit a Laplace-like resonance, but this one has mean period ratios 18:22:33 and mean orbital ratios 11:9:6. The equation is now: 3 × angle for Styx − 5 × angle for Nix + 2 × angle for Hydra = 180° Triple conjunctions are impossible, by the same reasoning as for the Jovian moons. There are five Styx/Hydra conjunctions and 3 Nix/Hydra conjunctions for every two Styx/Nix conjunctions. Europa, Ganymede, and Callisto all have icy surfaces. Several lines of evidence indicate that all three of them have oceans of liquid water beneath the ice. The first moon to be suspected of hosting such an ocean was Europa. Some heat source is needed to melt the ice. Tidal forces from Jupiter squeeze Europa repeatedly, but resonances with Io and Ganymede prevent it escaping by changing orbit. The squeezing heats the core of the moon, and calculations suggest that the amount of heat is enough to melt much of the ice. Since the surface is solid ice, the liquid water must be deeper down, probably forming a thick spherical shell. As further support, the surface is extensively cracked, with few signs of craters. The most likely explanation is that the ice forms a thick layer

floating on the ocean. Jupiter’s strong magnetic field induces a weaker one in Europa, and when the Galileo orbiter measured Europa’s magnetic field, mathematical analysis suggested that a substantial mass of conducting material must lie under Europa’s ice. The most plausible substance, given the data, is salty water. Europa’s surface has a number of regions of ‘chaos terrain’ where the ice is very irregular and jumbled. One such is Conamara Chaos, which appears to be formed from innumerable ice rafts that have been broken and moved. Others are Arran, Murias, Narbeth, and Rathmore Chaos. Similar formations occur on Earth in pack ice, floating on the sea, when a thaw sets in. In 2011 a team led by Britney Schmidt explained that chaotic terrain forms when ice sheets, lying above lensshaped lakes of liquid water, collapse. These lakes are nearer the surface than the ocean itself, perhaps only three kilometres underground.2 A depression of this kind called Thera Macula has an underlying lake with as much water as the North American Great Lakes. Europa’s lenticular lakes are closer to the surface than the main ocean. The best estimates right now are that aside from such lakes, the outer layer of ice is about 10–30 km thick, and the ocean is 100 km deep. If so, Europa’s ocean has twice the volume of all of Earth’s oceans combined. Based on similar evidence, Ganymede and Callisto also have subsurface oceans. Ganymede’s outer layer of ice is thicker, about 150 km, and the ocean beneath is again about 100 km deep. Callisto’s ocean probably lies the same distance under the ice, with an ocean 50–200 km deep. All of these figures are speculative, and differences in the chemistry, such as the presence of ammonia, would change them significantly.

Conamara Chaos on Europa. Enceladus, a moon of Saturn, is very cold, with a mean surface temperature of 75 K (about minus 200°C). You’d expect it not to show much activity, and so did astronomers, until Cassini discovered that it emits huge geysers of ice particles, water vapour, and sodium chloride, hundreds of kilometres high. Some of this material escapes altogether, and is believed to be the main source of Saturn’s E ring, which contains 6% sodium chloride. The rest falls back to the surface. The most plausible explanation, a salty underground ocean, was confirmed in 2015 by a mathematical analysis of seven years’ worth of data on tiny wobbles in the moon’s orientation (jargon: libration), measured by observing accurate positions of its craters.3 The moon wobbles through an angle of 0·12 degrees. This is too large to be consistent with a rigid connection between Enceladus’s core and its icy surface, and it indicates a global ocean rather than a more limited polar sea. The ice above is probably 30–40 km thick, and the ocean is 10 km deep – more than the average for Earth’s oceans. Seven of Saturn’s moons orbit just beyond the edge of the planet’s outer main ring, the A ring. They’re very small and their density is extremely low, suggesting they have internal voids. Several are shaped like flying

saucers, and some have smooth patchy surfaces. They are Pan, Daphnis, Atlas, Prometheus, Pandora, Janus, and Epimetheus. In 2010 Sébastian Charnoz, Julien Salmon, and Aurélien Crida4 analysed how the ring might evolve, along with hypothetical ‘test bodies’ at its edge, concluding that these moons have been spat out by the rings as material passes outside the Roche limit. The Roche limit is usually defined to be the distance inside which moons come to pieces under gravitational stress, but conversely it’s also the distance outside which rings become unstable, unless stabilised by other mechanisms such as shepherd moons. Saturn’s Roche limit (140,000 ± 2000 km) is just outside the edge of the A ring (136,775 km). Pan and Daphnis are just inside the Roche limit, the other five are just outside. Astronomers had long suspected that there must be a connection between the rings and these moons, because their radial distances are so close together. The A ring has a very sharp boundary, created by a 7:6 resonance with Janus, which prevents most of the ring material moving further out. This resonance is temporary; the rings ‘push’ Janus further out while themselves initially moving inwards a little to conserve angular momentum. As Janus continues to move out, the rings can spread outwards again, passing the Roche limit. The analysis supports this view, showing that some ring material can temporarily be pushed outside the Roche limit by viscous spreading – much as a blob of syrup on the kitchen table will slowly spread out and become thinner. Their method combines an analytic model of the test bodies with a numerical fluid-dynamics model of the rings. Continued viscous spreading causes the rings to spit out a succession of tiny moonlets, whose orbits resemble reality quite closely. Calculations indicate that these moonlets are aggregates of ice particles from the rings, loosely held together by their own gravity, explaining their low density and curious shapes. The results also shed light on a long-standing question: the age of the rings. One theory is that the rings formed from the collapsing solar nebula at much the same time as Saturn did. However, a moonlet such as Janus should take no more than a hundred million years to drift outwards from the A ring to its current orbit, suggesting an alternative theory: both the rings and these moonlets appeared together when a larger moon passed inside the Roche limit and broke up, some tens of millions of years ago. The

simulations reduce this period to between 1 and 10 million years: the authors say, ‘Saturn’s rings, like a mini protoplanetary disk, may be the last place where accretion was recently active in the solar system, some 106–107 years ago.’

8 Off on a Comet It may be said with perfect truth that a fisherman standing on the sun’s surface, holding a rod long enough, could fling his line in no direction without hooking plenty of comets. Jules Verne, Off on a Comet! ‘WHEN BEGGARS DIE, THERE are no comets seen; the heavens themselves blaze forth the death of princes.’ So says Calpurnia in Act 2, Scene 2 of Shakespeare’s Julius Caesar, prophesying Caesar’s demise. Of the five references to comets in Shakespeare, three reflect the ancient belief that comets are harbingers of disaster. These strange and puzzling objects appear unexpectedly in the night sky trailing a bright curved tail, move slowly across the background of stars, and disappear again. They are unheralded interlopers that don’t seem to fit the normal patterns of celestial events. Reasonable enough, then, in times when none knew better, and when priests and shamans were always seeking to bolster their influence, to interpret a comet as a messenger from the gods. The common assumption was that the message was ominous. There were enough natural disasters around that if that’s what you wanted to believe, it wasn’t hard to find convincing confirmation. Comet McNaught, which appeared in 2007, was the brightest for 40 years. Clearly it heralded the 2007–8 financial crisis. See? Anyone can do it. Priests claimed to know what comets were for, but neither they nor philosophers knew where they were located. Were they celestial bodies, like the stars and planets? Or were they meteorological phenomena, like clouds? The looked a bit like clouds; they were fuzzy, not crisp like stars and planets. But they moved more like planets, except for those sudden appearances and disappearances. Ultimately, the debate was settled by scientific evidence. When the astronomer Tycho Brahe used precision measurements to estimate the distance to the Great Comet of 1577, he

demonstrated that it was further away than the Moon. Since clouds hide the Moon but not the other way round, comets are of celestial habitat. Great Comet of 1577 over Prague. Engraving by Jiri Daschitzky. By 1705 Edmond Halley had gone further, showing that at least one comet is a regular visitor to the night skies. Comets are like planets: they orbit the Sun. They seem to disappear when they get too far away to see, and reappear when they get close enough again. Why do they grow tails, and lose them again? Halley wasn’t sure, but it was related to their proximity to the Sun. Halley’s insight into comets was one of the first big discoveries in astronomy to be deduced from the mathematical patterns discovered by Kepler and reinterpreted more generally by Newton. Since planets move in ellipses, Halley reasoned, why not comets too? If so, their motion would be periodic, and the same comet would return repeatedly to Earthly skies at equally spaced times. Newton’s law of gravity modified this statement a little: the motion would be almost periodic, but the gravitational attraction of other planets, especially the giants Jupiter and Saturn, would accelerate or retard the comet’s return. To test this theory, Halley delved into arcane records of comet sightings. Before Galileo’s invention of the telescope, only comets visible to the naked eye could be seen. A few were unusually bright, with an impressive

tail. Petrus Apianus saw one in 1531; Kepler had observed another in 1607; a similar comet had appeared during Halley’s lifetime, in 1682. The gaps between these dates are 76 and 75 years. Might all three sightings be of the same body? Halley was convinced they were, and he predicted that the comet would return in 1758. He was right, just. On Christmas day of that year the German amateur astronomer Johann Palitzsch saw a faint smudge in the sky, which soon developed the characteristic tail. By then, three French mathematicians, Alexis Clairaut, Joseph Lalande, and Nicole-Reine Lepaute, had carried out more accurate calculations, amending the predicted date for the comet’s closest approach to the Sun to 13 April. The actual date was a month earlier, so perturbations by Jupiter and Saturn had delayed the comet by 618 days. Halley died before his prediction could be verified. What we now call Halley’s comet (named after him in 1759) was the first body other than a planet to be shown to orbit the Sun. By comparing ancient records to modern computations of its past orbit, Halley’s comet can be traced back to 240 BC, when it was seen in China. Its next appearance in 164 BC was noted on a Babylonian clay tablet. The Chinese saw it again in 87 BC, 12 BC, 66 AD, 141 AD, … and so on. Halley’s forecast of the comet’s eventual return was one of the earliest truly novel astronomical predictions to be based on a mathematical theory of celestial dynamics. Comets aren’t just an abstruse astronomical puzzle. In the introduction I mentioned a far-reaching theory involving comets: for the past few decades they’ve been the preferred explanation of how the Earth got its oceans. Comets are mainly composed of ice; the tail forms when the comet gets close enough to the sun for the ice to ‘sublimate’, that is, turn directly from solid to vapour. There’s convincing circumstantial evidence that lots of comets collided with the early Earth, in which case the ice would melt and collect to form oceans. Water would also be incorporated into the molecular structure of crustal rocks, which actually contain quite a lot of it. Earth’s water is vital to the planet’s life forms, so understanding comets has the potential to tell us something important about ourselves. Alexander Pope’s 1734 poem An Essay on Man includes the memorable line ‘the

proper study of mankind is Man’. However, without going into the poem’s spiritual and ethical intentions, any study of humanity should also involve the context for human beings, not just the beings themselves. That context is the entire universe – so, Pope’s dictum notwithstanding, the proper study of mankind is everything. Today, astronomers have catalogued 5253 comets. There are two main types: long-period comets with periods 200 years or greater, whose orbits extend beyond the outer reaches of the solar system, and short-period comets that stay closer to the Sun and often have rounder, though still elliptical, orbits. Halley’s comet, with its 75-year period, is a short-period comet. A few comets have hyperbolic orbits: we’ve already encountered the hyperbola, another conic section familiar to ancient Greek geometers, on page 17. Unlike an ellipse, it doesn’t close up. Bodies following such an orbit appear from a vast distance, swing past the Sun, and if they manage not to collide with it they head back out into space, never to be seen again. A hyperbolic shape suggests these comets fall Sunwards from interstellar space, but astronomers now think that most of them, perhaps all, originally followed very distant closed orbits before being perturbed by Jupiter. The distinction between ellipses and hyperbolas involves the energy of the body. Below a critical value of the energy, the orbit is a closed ellipse. Above that value, it’s a hyperbola. At that value, it’s a parabola. A comet in a very large elliptical orbit, perturbed by Jupiter, gains extra energy, which can be enough to tip it past the critical value. A close encounter with an outer planet can add more energy through the slingshot effect: the comet steals some of the planet’s energy, but the planet is so massive that it doesn’t notice. In this manner, the orbit can become a hyperbola. A parabolic orbit is unlikely because it’s poised at the critical energy value. But, for just that reason, a parabola was often used as a first step towards computing a comet’s orbital elements. A parabola is close to both an ellipse and a hyperbola. This brings us back to the short-period comet that hit the headlines, named 67P/Churyumov–Gerasimenko after its discoverers Klim Churyumov and

Svetlana Gerasimenko. It orbits the Sun every six and a half years. 67P’s hitherto commonplace cometary existence, pottering round the Sun and expelling jets of heated water vapour when it got too close, came to the attention of astronomers, and the Rosetta spacecraft was sent to rendezvous with it. As Rosetta approached its goal, 67P was revealed as a cosmic rubber duck: two round lumps joined by a narrow neck. At first no one was sure whether this shape arose from two rounded bodies that came together very slowly, or a single body that eroded in the neck region. Late in 2015 this issue was resolved by an ingenious application of mathematics to detailed images of the comet. At first sight 67P’s terrain seems jumbled and irregular, with jagged cliffs and flat depressions assembled at random, but its surface detail provides clues to its origins. Imagine taking an onion, slicing off random pieces, and hacking chunks out of it. Thin slices parallel to the surface would leave flat spots, deeper gouges would show a stack of separate layers. The comet’s flat depressions are akin to the slices, and its cliffs and other regions often show layered strata of ice. Series of layers can be seen at the top and centre right of the image on page 2, for example, and many flat regions are visible. Astronomers think that when comets first appeared in the early solar system they grew by accretion, so that layer upon layer of ice was gradually added, much like the layers of an onion. So we can ask whether the geological formations visible in images of 67P are consistent with this theory, and if so, we can use the geology to reconstruct the comet’s history. Matteo Massironi and coworkers carried out this task in 2015.1 Their results offer strong support for the theory that the duck shape was created by a gentle collision. The basic idea is that the comet’s history can be deduced from the geometry of its ice layers. Eyeballing the images, the two-body theory looks a better bet, but Massironi’s team carried out a careful mathematical analysis using three-dimensional geometry, statistics, and mathematical models of the comet’s gravitational field. Starting from a mathematical representation of the observed shape of the comet’s surface, the team first worked out the positions and orientations of 103 planes, each providing the best fit to a geological feature associated with the observed layers, such as a terrace (flat region) or cuesta (a type of ridge). They found that these planes fit together consistently around each lobe, but not at the

neck where the lobes join. This indicates that each lobe acquired onion-like layers as it grew, before they came together and stuck. Two competing scenarios for the structure of 67P. Left: Collision theory. Right: Erosion theory. Schematic illustration of best-fitting planes to terraces and cuestas. Left: Collision theory. Right: Erosion theory. The actual calculation was performed in three dimensions using a statistical measure for the best fit, and used 103 planes. When the layers form, they’re roughly perpendicular to the local direction of gravity – a technical way to say that the additional material falls downwards. So for further confirmation, the team computed the gravitational field of the comet under each of the two hypotheses, and used statistical methods to show that the layers fit the collision model better. Despite being made mainly of ice, 67P is black as midnight and pockmarked with thousands of rocks. Philae made a difficult, and as it turned out, temporary, landing on the duck’s head. The landing did not go as intended. Philae’s equipment included a small rocket motor, spikes with screw threads, harpoons, and a solar panel. The plan was to make a gentle landing, fire the rocket to keep the lander pressed against the comet’s surface, harpoon the comet to hold it in place when the rockets were turned off, screw the spikes into the comet to make sure it stayed there, and then use the solar panel to harvest energy from sunlight. Men, mice, plans, bestlaid … the rocket failed to fire, the harpoons failed to stick, the screws

failed to bite, and as a result the solar panel ended up in deep shadow with hardly any sunlight to harvest. Despite its proverbial ‘perfect three-point landing – two knees and a nose’, Philae attained almost all of its scientific objectives, sending back vital data. It was hoped that it might add more as the comet got closer to the Sun, the light got stronger, and the probe woke up from its electronic sleep. Philae did briefly renew contact with the ESA, but communication was lost again, probably because the increasing activity of the comet damaged it. Before it ran out of power, Philae confirmed that the comet’s surface is ice with a coating of black dust. As already mentioned, it also sent measurements showing that the ice contains a greater proportion of deuterium than Earth’s oceans, casting serious doubt on the theory that the water in the oceans was mainly delivered by comets when the solar system was forming. Ingenious work using the data that did make it home has provided further useful information. For example, mathematical analysis of how Philae’s landing struts compressed shows that in places 67P has a hard crust, but elsewhere the surface is softer. Images made by Rosetta include three marks where the lander first hit the comet, deep enough to show that the material there is relatively soft. Philae’s on-board hammer was unable to penetrate the ice where it came to rest, so there the ground is hard. On the other hand, the bulk of 67P is very porous: three quarters of its interior is empty space. Philae sent back some intriguing chemistry, too: several simple organic (this means carbon-based, and is not indicative of life) compounds, and a more complex one, polyoxymethylene, probably created from the simpler molecule formaldehyde by the action of sunlight. Astronomers were startled by one of Rosetta’s chemical discoveries: quite a lot of oxygen molecules in the gas cloud surrounding the comet.2 They were so surprised that at first they assumed they’d made a mistake. In conventional theories of the origin of the solar system, the oxygen would have been heated, making it react with other elements to form compounds such as carbon dioxide, so it would no longer be around as pure oxygen. The early solar system must have been less violent than previously thought, allowing grains of solid oxygen to build up slowly and avoid forming compounds.

That doesn’t conflict with the more dramatic events that are thought to have occurred during the formation of the solar system, such as migrating planets and colliding planetesimals, but it suggests that such events must have been relatively rare, punctuating a background of slow, gentle growth. Where do comets come from? Long-period comets can’t hang around indefinitely in their present orbits. As they pass through the solar system, there’s a risk of a collision, or a close encounter that hurls them off into space, never to return. The chance may be small, but over millions of orbits the odds against avoiding such disasters mount up. Moreover, comets decay, losing mass every time they round the Sun and stream off sublimating ice. Hang around too long, and they melt away. In 1932 Ernst Öpik suggested a way out: there must be a huge reservoir of icy planetesimals in the outer reaches of the solar system, which replenishes the supply of comets. Jan Oort had the same idea independently in 1950. From time to time one of these icy bodies is dislodged, perhaps by near misses with another one, or just by chaotic gravitational perturbations. It then changes its orbit, falling in towards the Sun, warms up, and the characteristic coma and tail are born. Oort investigated this mechanism in considerable mathematical detail, and in his honour we now name the source the Oort cloud. (As explained earlier for asteroids, the name should not be taken literally. It’s a very sparse cloud.) The Oort cloud is thought to occupy a vast region round the Sun between about 5000 AU out to 50,000 AU (0·03 to 0·79 light years). The inner cloud, out to 20,000 AU, is a torus roughly aligned with the ecliptic; the outer halo is a spherical shell. There are trillions of bodies a kilometre or more across in the outer halo, and the inner cloud contains a hundred times as many. The total mass of the Oort cloud is about five times that of the Earth. This structure has not been observed: it’s deduced from theoretical calculations. Simulations and other evidence suggest that the Oort cloud came into existence when the local protoplanetary disc began to collapse and form the solar system. We’ve discussed evidence that the resulting planetesimals

were originally closer to the Sun, and were then hurled into the outermost regions by the giant planets. The Oort cloud could be a remnant of the early solar system formed from leftover debris. Alternatively, it may be the result of a competition between the Sun and neighbouring stars to attract material that was always that far out, near the borderline where the two stars’ gravitational fields cancel each other. Or, as proposed in 2010 by Harold Levison and coworkers, the Sun stole debris from the protoplanetary discs of the cluster of 200 or so stars in its vicinity. If the ejection theory is correct, the initial orbits of the bodies in the Oort cloud were very long, thin ellipses. However, since these bodies mostly stay in the cloud, their orbits must now be much fatter, almost circular. It’s thought that the orbits were fattened up by interaction with nearby stars and galactic tides – the overall gravitational effect of the Galaxy. Short-period comets are different, and are thought to have a different origin: the Kuiper belt and the scattered disc. When Pluto was discovered, and found to be quite small, many astronomers wondered if it might be another Ceres – the first new body in a huge belt containing thousands. One – not the first – was Kenneth Edgeworth, who suggested in 1943 that when the outer solar system past Neptune condensed from the primal gas cloud, the matter wasn’t dense enough to form large planets. He also saw these bodies as a potential source of comets. In 1951 Gerald Kuiper proposed that a disc of small bodies might have collected in that region early in the formation of the solar system, but he thought (as many then did) that Pluto was about the size of the Earth, so it would have disturbed the disc and scattered its contents far and wide. When it turned out that such a disc still exists, Kuiper received the dubious honour of having an astronomical region named after him because he did not predict it. Several individual bodies were discovered in this region: we’ve encountered them already as TNOs. What clinched the existence of the Kuiper belt was, once more, comets. In 1980 Julio Fernández carried out a

statistical study of short-period comets. There are too many for them all to have come from the Oort cloud. Out of every 600 comets emanating from the Oort cloud, 599 would become long-period comets and only one would be captured by a giant planet and change to a short-period orbit. Perhaps, Fernández said, there’s a reservoir of icy bodies between 35 and 50 AU from the Sun. His ideas received strong support from a series of simulations carried out by Martin Duncan, Tom Quinn, and Scott Tremaine in 1988, who also noted that shortperiod comets stay close to the ecliptic, but longperiod ones arrive from almost any direction. The proposal became accepted, with the name ‘Kuiper belt’. Some astronomers prefer ‘Edgeworth–Kuiper belt’ and others assign credit to neither. The origins of the Kuiper belt are murky. Simulations of the early solar system indicate the scenario mentioned earlier, in which the four giant planets originally formed in a different order (reading outwards from the Sun) from today’s, and then migrated, scattering planetesimals to the four winds. Most of the primordial Kuiper belt was flung away, but one body in a hundred remained. Like the inner region of the Oort cloud, the Kuiper belt is a fuzzy torus. The distribution of matter in the Kuiper belt isn’t uniform; like the asteroid belt, it’s modified by resonances, in this case with Neptune. There’s a Kuiper cliff at about 50 AU, where the number of bodies falls off suddenly. This has not been explained, although Patryk Lyakawa speculates that it might be the result of an undetected large body – a genuine Planet X. The scattered disc is even more enigmatic and less well known. It overlaps the Kuiper belt slightly, but it extends further, to about 100 AU, and is strongly inclined relative to the ecliptic. Bodies in the scattered disc have highly elliptical orbits and are often diverted into the inner solar system. There they linger for a time as centaurs, before the orbit changes again and they turn into short-period comets. Centaurs are bodies occupying orbits that cross the ecliptic between the orbits of Jupiter and Neptune; they persist for only a few million years, and there are probably about 45,000 of them more than a kilometre across. The majority of short-period comets probably come from the scattered disc rather than the Kuiper belt.

In 1993 Carolyn and Eugene Shoemaker and David Levy discovered a new comet, later named Shoemaker–Levy 9. Unusually, it had been captured by Jupiter and was in orbit around the giant planet. Analysis of its orbit indicated that the capture had occurred 20–30 years earlier. Shoemaker– Levy 9 was unusual in two ways. It was the only comet observed orbiting a planet, and it seemed to have broken into pieces. Shoemaker-Levy 9 on 17 May 1994. The reason emerged from a simulation of its orbit. Calculating backwards, in 1992 the comet must have passed inside Jupiter’s Roche limit. Gravity’s tidal forces then broke the comet up, creating a string of about 20 fragments. The comet had been captured by Jupiter around 1960– 70, and the close encounter had distorted its orbit into a long, thin one. Simulating the orbit in forward time predicted that on the comet’s next fly-by, in July 1994, it would collide with Jupiter. Astronomers had never observed a celestial collision before, so this discovery caused considerable excitement. A collision would stir up Jupiter’s atmosphere, making it possible to find out more about its deeper layers, normally hidden by the cloud above. In the event, the impact was even more dramatic than expected, leaving a chain of gigantic scars across the planet that slowly faded, remaining visible for months. Twenty-one impacts in all were sighted; the largest produced 600 times as much energy as all the nuclear weapons on Earth if they were exploded simultaneously.

Dark spots are some of the impact sites from fragments of Shoemaker–Levy 9. The impacts taught scientists a lot of new things about Jupiter. One is its role as a celestial hoover. Shoemaker–Levy 9 may have been the only comet observed orbiting Jupiter, but at least five others must have done so in the past, judging by their current orbits. All such captures are temporary: either the comet is recaptured by the Sun or it eventually collides with something. Thirteen chains of craters on Callisto and three on Ganymede suggest that sometimes what it hits isn’t Jupiter. Put together, the evidence shows that Jupiter sweeps up comets and other cosmic debris, by capturing them and then colliding with them. Such events are rare by our standards, but frequent on a cosmic timescale: a comet 1·6 km across hits Jupiter every 6000 years or so, with smaller ones colliding more often This aspect of Jupiter helps to protect the inner planets from comet and asteroid impacts, leading to the suggestion in Peter Ward and Donald Brownlee’s Rare Earth3 that a large planet like Jupiter makes its inner worlds more habitable for life. Unfortunately for this seductive line of reasoning, Jupiter also disturbs asteroids from the main belt and these can collide with inner planets. If Jupiter were slightly smaller, its overall effect would be detrimental to life on Earth.4 At its current size, there seems to be no significant overall advantage for Earthly life. Rare Earth is ambivalent about impacts in any case: it hails Jupiter as our saviour from comets, while praising its tendency to fling asteroids around as a way to shake up ecosystems and encourage more rapid evolution.

Shoemaker–Levy 9 brought home to many American congressmen the extraordinary violence of a comet impact. The largest impact scar on Jupiter was the same size as the Earth. There’s no way we could protect ourselves against an impact of this magnitude with current or foreseeable technology, but it did rather focus the mind on lesser impacts, be they from a comet or an asteroid, where we might be able to prevent a collision if we took steps to give ourselves enough prior warning. Congress rapidly instructed NASA to catalogue all near-Earth asteroids more than a kilometre across. So far 872 have been detected, of which 153 might potentially hit us. Estimates suggest another 70 or so exist, but haven’t yet been spotted.

9 Chaos in the Cosmos This is highly irregular. Airplane II: The Sequel PLUTO’S MOONS ARE WOBBLY. Pluto has five satellites. Charon is spherical and unusually large compared to its primary, while Nix, Hydra, Kerberos, and Styx are tiny irregular lumps. Charon and Pluto are tidally locked so that each presents the same face to the other. Not so the other moons. In 2015, the Hubble telescope observed irregular variations in the light reflected from Nix and Hydra. Using a mathematical model of spinning bodies, astronomers deduced that these two moons must be tumbling end over end, but not in a nice regular way. Instead, their motion is chaotic.1 In mathematics, ‘chaotic’ is not just a fashionable word for ‘erratic and unpredictable’. It refers to deterministic chaos, which is apparently irregular behaviour resulting from entirely regular laws. That probably sounds paradoxical, but the combination is often unavoidable. Chaos looks random – and in certain respects it is – but it stems from the same mathematical laws that produce regular, predictable behaviour like the Sun rising every morning. Further Hubble measurements suggest that Styx and Kerberos also spin chaotically. One of the tasks carried out by New Horizons when it visited Pluto was to verify this theory. Its data were to be transmitted back to Earth over a 16-month period, and as I write, the results haven’t yet arrived. Pluto’s wobbly moons are the breaking news on chaotic dynamics in the cosmos, but astronomers have discovered many examples of cosmic chaos, from fine details about tiny moons to the long-term future of the solar system. Saturn’s moon Hyperion is another chaotic tumbler – the first satellite to be caught behaving badly. The Earth’s axis is tilted at a fairly stable 23·4 degrees, giving us the regular succession of seasons, but the

axial tilt of Mars varies chaotically. Mercury and Venus used to be like that too, but tidal effects from the Sun have stabilised them. There’s a link between chaos and the 3:1 Kirkwood gap in the asteroid belt. Jupiter clears out asteroids from this region, flinging them willy-nilly around the solar system. Some cross the orbit of Mars, which can redirect them almost anywhere. Maybe that’s why the dinosaurs met their demise. Jupiter’s Trojan asteroids were probably captured as a consequence of chaotic dynamics. Chaotic dynamics has even provided a way for astronomers to estimate the age of a family of asteroids. Far from being a gigantic clockwork machine, the solar system plays roulette with its planets. The first hint along these lines, found by Gerry Sussman and Jack Wisdom in 1988, was the discovery that Pluto’s orbital elements vary erratically as a consequence of the gravitational forces exerted on it by the other planets. A year later, Wisdom and Laskar showed that the Earth’s orbit is also chaotic, though in a milder way: the orbit itself doesn’t greatly change, but the Earth’s position along the orbit is unpredictable in the long term – 100 million years from now. Sussman and Wisdom also showed that if there were no inner planets, Jupiter, Saturn, Uranus, and Neptune would behave chaotically in the long term. These outer planets have a significant effect on all the other planets, making them the main source of chaos in the solar system. However, chaos isn’t confined to our celestial backyard. Calculations indicate that many exoplanets around far stars probably follow chaotic orbits. There is astrophysical chaos: the light output of some stars varies chaotically. 2 The motion of stars in galaxies may well be chaotic, even though astronomers usually model their orbits as circles (see Chapter 12). Chaos, it seems, rules the cosmos. Yet astronomers have found that, more often than not, the main cause of chaos is resonant orbits, simple numerical patterns. Like that 3:1 Kirkwood gap. On the other hand, chaos is also responsible for patterns – the spirals of galaxies may well be an example, as we’ll also see in Chapter 12. Order creates chaos, and chaos creates order.

Random systems have no memory. If you roll a dice3 twice, the number that turns up on the first throw tells you nothing about what will happen on the second throw. It might be the same number as before; then again, it might not be. Don’t believe anyone who tries to tell you that if a dice hasn’t thrown a 6 for a long time, then the ‘law of averages’ makes a 6 more likely. There’s no such law. It’s true that in the long run the proportion of 6s for fair dice should be very close to 1/6, but that happens because large numbers of new tosses swamp any discrepancies, not by the dice suddenly deciding to catch up to where a theoretical average says they ought to be.4 Chaotic systems, in contrast, do have a kind of short-term memory. What they’re doing now provides hints about what they will do a little into the future. Ironically, if dice were chaotic, then not having thrown a 6 for a long time would be evidence that it probably won’t happen soon.5 Chaotic systems have lots of approximate repetitions in their behaviour, so the past is a reasonable – though far from foolproof – guide to the near future. The length of time for which this kind of forecasting remains valid is called the prediction horizon (jargon: Liapunov time). The more accurately you know the current state of a chaotic dynamical system, the longer the prediction horizon becomes – but the horizon increases far more slowly than the precision of the measurements. However precise they are, the slightest error in the current state eventually grows so large that it overwhelms the prediction. The meteorologist Edward Lorenz discovered this behaviour in a simple model motivated by weather, and the same is true of the sophisticated weather models used by forecasters. The movement of the atmosphere obeys specific mathematical rules with no element of randomness, yet we all know how unreliable weather forecasts can become after just a few days. This is Lorenz’s famous (and widely misunderstood) butterfly effect: a flap of a butterfly’s wing can cause a hurricane a month later, halfway round the world.6 If you think that sounds implausible, I don’t blame you. It’s true, but only in a very special sense. The main potential source of misunderstanding is the word ‘cause’. It’s hard to see how the tiny amount of energy in the flap of a wing can create the huge energy in a hurricane. The answer is, it doesn’t. The energy in the hurricane doesn’t come from the flap: it’s

redistributed from elsewhere, when the flap interacts with the rest of the otherwise unchanged weather system. After the flap, we don’t get exactly the same weather as before except for an extra hurricane. Instead, the entire pattern of weather changes, worldwide. At first the change is small, but it grows – not in energy, but in difference from what it would otherwise have been. And that difference rapidly becomes large and unpredictable. If the butterfly had flapped its wings two seconds later, it might have ‘caused’ a tornado in the Philippines instead, compensated for by snowstorms over Siberia. Or a month of settled weather in the Sahara, for that matter. Mathematicians call this effect ‘sensitive dependence on initial conditions’. In a chaotic system, inputs that differ very slightly lead to outputs that differ by large amounts. This effect is real, and very common. For example, it’s why kneading dough mixes the ingredients thoroughly. Every time the dough is stretched, nearby grains of flour move further apart. When the dough is then folded over to stop it escaping from the kitchen, grains that are far apart may (or may not) end up close together. Local stretching, combined with folding, creates chaos. That’s not just a metaphor: it’s a description in ordinary language of the basic mathematical mechanism that generates chaotic dynamics. Mathematically, the atmosphere is like the dough. The physical laws that govern the weather ‘stretch’ the state of the atmosphere locally, but the atmosphere doesn’t escape from the planet, so its state ‘folds back’ on itself. Therefore, if we could run the Earth’s weather twice, with the only difference being an initial flap or no-flap, the resulting behaviours would diverge exponentially. The weather would still look like weather, but it would be different weather. In reality we can’t run real weather twice, but this is precisely what happens in weather forecasts using models that reflect genuine atmospheric physics. Very tiny changes to the numbers representing the current state of the weather, when input into the equations that predict the future state, lead to large-scale changes in the forecast. For example, an area of high pressure over London in one simulation can be replaced by an area of low pressure in another. The current way round this annoying effect is to run many simulations with small random variations in initial conditions, and use the

results to quantify how probable different predictions are. That’s what ‘20% chance of thunderstorms’ means. In practice it’s not possible to cause a specific hurricane by employing a suitably trained butterfly, because forecasting the effect of the flap is also subject to the same prediction horizon. Nevertheless, in other contexts, such as the heartbeat, this kind of ‘chaotic control’ can provide an efficient route to desired dynamic behaviour. We’ll see several astronomical examples in Chapter 10, in the context of space missions. Not convinced? A recent discovery about the early solar system puts the issue into sharp relief. Suppose some celestial superpower could run the formation of the solar system from a primordial gas cloud again, using exactly the same initial state except for one extra molecule of gas. How different would today’s solar system be? Not a lot, you might think. But remember the butterfly effect. Mathematicians have proved that bouncing molecules in a gas are chaotic, so it wouldn’t be a surprise if the same were true of collapsing gas clouds, even though the details are technically different. To find out, Volker Hoffmann and coworkers simulated the dynamics of a disc of gas at a stage when it contains 2000 planetesimals, keeping track of how collisions cause these bodies to aggregate into planets.7 They compared the results with simulations including two gas giants, with two distinct choices for their orbits. They made a dozen runs for each of these three scenarios, with slightly different initial conditions. Each run took about a month on a supercomputer. They found that planetesimal collisions are chaotic, as expected. The butterfly effect is dramatic: change the initial position of a single planetesimal by just one millimetre, and you get a completely different system of planets. Extrapolating from this result, Hoffmann thinks that by adding a single molecule of gas to an exact model of the nascent solar system (were such a thing possible) you’d change the outcome so much that the Earth fails to form. So much for the clockwork universe.

Before we get carried away by how incredibly unlikely this makes our existence, and invoking the divine hand of providence, we should take into account another aspect of the calculations. Although each run leads to planets with different sizes and different orbits, all of the solar systems that arise for a given scenario are very similar to each other. Without any gas giants, we get about 11 rocky worlds, most of them smaller than the Earth. Add the gas giants – a more realistic model – and we get four rocky planets, with masses between half that of the Earth and a bit more than that of the Earth. That’s very close to what we actually have. Although the butterfly effect changes orbital elements, the overall structure is much the same as before. The same happens in weather models. Flap … and the global weather is different from what it would have been – but it’s still weather. You don’t suddenly get floods of liquid nitrogen or a blizzard of giant frogs. So although our solar system would not have arisen in exactly its present form if the initial gas cloud had been the slightest bit different, something remarkably similar would have arisen instead. So living organisms would probably have been just as likely to evolve. The prediction horizon can sometimes be used to estimate the age of a chaotic system of celestial bodies, because it governs how fast the system breaks up and disperses. Asteroid families are examples. They can be spotted because their members have very similar orbital elements. Each family is thought to have been created by the break-up of a single larger body at some time in the past. In 1994 Andrea Milani and Paolo Farinella used this method to deduce that the Veritas asteroid family is at most 50 million years old.8 This is a compact cluster of asteroids associated with 490 Veritas, towards the outside of the main belt and just inside the 2:1 resonant orbit with Jupiter. Their calculations show that two of the asteroids in this family have strongly chaotic orbits, created by a temporary 21:10 resonance with Jupiter. The prediction horizon implies that these two asteroids should not have stayed close together for more than 50 million years, and other evidence suggests they are both original members of the Veritas family.

The first person to recognise the existence of deterministic chaos, and to gain some inkling of why it happens, was the great mathematician Henri Poincaré. He was competing for a mathematical prize offered by King Oscar II of Norway and Sweden, asking for a solution of the n-body problem for Newtonian gravitation. The rules for the prize specified what sort of solution was required. Not a formula like Kepler’s ellipse, because everyone was convinced no such thing existed, but ‘a representation of the coordinates of each point as [an infinite] series in a variable that is some known function of time and for all of whose values the series converges uniformly’. Poincaré discovered that the task is essentially impossible, even for three bodies under very restrictive conditions. The way he proved it was to demonstrate that orbits can be what we now call ‘chaotic’. The general problem for any number of bodies proved too much even for Poincaré. He took n = 3. In fact, he worked on what I called the 2½- body problem in Chapter 5. The two bodies are, say, a planet and one of its moons; the half-body is a grain of dust, so lightweight that although it responds to the gravitational fields of the other two bodies, it has absolutely no effect on them. What emerges from this model is a lovely combination of perfectly regular two-body dynamics for the massive bodies and highly erratic behaviour for the dust particle. Ironically, it’s the regular behaviour of the massive bodies that makes the dust particle go crazy. ‘Chaos’ makes it sound as though the orbits of three or more bodies are random, structureless, unpredictable, and lawless. Actually, the dust particle loops round and round in smooth paths close to arcs of ellipses, but the shape of the ellipse keeps changing without any obvious pattern. Poincaré came across the possibility of chaos when he was thinking about the dynamics of the dust grain when it happens to be close to a periodic orbit. What he expected was some complicated combination of periodic motions with different periods, much as an orbiting capsule goes round the Moon goes round the Earth goes round the Sun – all in different periods of time. However, as already specified in the rules for the prize, the answer was expected to be a ‘series’, which combines infinitely many periodic motions, not just three. Poincaré found such a series. How, then, does chaos appear? Not as a consequence of the series, but because of a flaw in the whole idea. The

rules stated that the series must converge. This is a technical mathematical requirement for an infinite sum to make sense. Essentially, the sum of the series should get closer and closer to some specific number as you include more and more terms. Poincaré was alert to pitfalls, and realised that his series didn’t converge. At first, it seemed to be getting closer and closer to some specific number, but then the sum started to diverge from that number by ever greater amounts. This behaviour is characteristic of an ‘asymptotic’ series. Sometimes an asymptotic series is useful for practical purposes, but here it hinted at an obstacle to obtaining a genuine solution. To figure out what that obstacle was, Poincaré abandoned formulas and series, and turned to geometry. He considered both position and velocity, so the contour lines in the picture on page 79 are really three-dimensional objects, not curves. This causes extra complications. When he thought about the geometric arrangement of all the possible orbits near a particular periodic one, he realised that many orbits must be very tangled and erratic. The reason lay in a special pair of curves, which captured how nearby orbits either approach the periodic one or diverge from it. If these curves cross each other at some point, then basic mathematical features of dynamics (uniqueness of solutions of a differential equation for given initial conditions) imply that they must cross at infinitely many points, forming a tangled web. Soon after, in Les Méthodes Nouvelles de la Mécanique Celeste (New Methods of Celestial Mechanics) he described the geometry as: a kind of trellis, a fabric, a network of infinitely tight mesh; each of the two curves must not cross itself but it must fold on itself in a very complicated way to intersect all of the meshes of the fabric infinitely many times. One will be struck by the complexity of this picture, which I will not even attempt to draw. Today we call this picture a homoclinic tangle. Ignore ‘homoclinic’ (jargon: an orbit that joins an equilibrium point to itself) and focus on ‘tangle’, which is more evocative. The picture on page 127 explains the geometry in a simple analogue. Ironically, Poincaré very nearly missed making this epic discovery. While looking through documents in the Mittag-Leffler Institute in Oslo, the mathematical historian June Barrow-Green discovered that the published version of his prizewinning work was not the one he’d

submitted.9 After the prize had been awarded and the official memoir had been printed but not yet distributed, Poincaré had discovered a mistake – he’d overlooked chaotic orbits. He withdrew his memoir and paid for a revised ‘official’ version to be quietly substituted. It took a while for Poincaré’s new ideas to sink in. The next big advance came in 1913 when George Birkhoff proved the ‘Last Geometric Theorem’, an unproved conjecture that Poincaré had used to deduce the occurrence, in suitable circumstances, of periodic orbits. We now call this result the Poincaré–Birkhoff fixed point theorem. Mathematicians and other scientists became fully aware of chaos about fifty years ago. Following in Birkhoff’s footsteps, Stephen Smale made a deeper study of the geometry of the homoclinic tangle, having encountered the same problem in another area of dynamics. He invented a dynamical system with much the same geometry that’s easier to analyse, known as the Smale horseshoe. This system starts with a square, stretches it out into a long thin rectangle, folds it round in a horseshoe shape, and fits it back on top of the original square. Repeating this transformation is much like kneading dough, and it has the same chaotic consequences. The horseshoe geometry allows a rigorous proof that this system is chaotic, and that in some respects it behaves like a random sequence of coin tosses – despite being completely deterministic.

Smale’s horseshoe. The square is repeatedly folded, creating a series of horizontal stripes. Reversing time and unfolding it converts these into similar vertical stripes. When the two sets of stripes cross, we get a homoclinic tangle. The dynamics – obtained by repeatedly folding – makes points jump around on the tangle, apparently at random. The complete tangle involves infinitely many lines. As the extent and richness of chaotic dynamics became apparent, the growing excitement triggered a lot of interest from the media, who dubbed the whole enterprise ‘chaos theory’. Really, the topic is one part – a significant and fascinating part, to be sure – of an even more important area of mathematics, known as nonlinear dynamics. The strange behaviour of Pluto’s moons is just one example of chaos in the cosmos. In 2015 Mark Showalter and Douglas Hamilton published a mathematical analysis backing up the Hubble’s puzzling observations of the moons of Pluto.10 The idea is that Pluto and Charon act like the dominant bodies in Poincaré’s analysis, and the other, far smaller, moons act a bit like the dust particle. However, because they’re not point particles, but shaped like rugby footballs, or possibly even potatoes, their craziness shows up as chaotic tumbling. Their orbits, and where the moons will be in those orbits

at any given time, are also chaotic: predictable only statistically. Even less predictable is the direction in which each moon will point. Pluto’s moons weren’t the first tumblers to be spotted. That honour goes to Saturn’s satellite Hyperion, and at the time it was thought to be the only tumbling moon. In 1984 Hyperion attracted the attention of Wisdom, Stanton Peale, and François Mignard.11 Almost all moons in the solar system fall into two categories. The axial spin of a moon in the first category has been heavily modified by tidal interactions with its parent planet, so it always presents the same face to the planet, a 1:1 spin–orbital resonance, otherwise known as synchronous rotation. For the second category, very little interaction has taken place and it still spins much as it did when it first formed. Hyperion and Iapetus are exceptions: according to theory, they should eventually lose most of their initial spin and synchronise it with their orbital revolution, but not for a long time – about a billion years. Despite that, Iapetus already rotates synchronously. Hyperion alone seemed to be doing something more interesting. The question was: what? Wisdom and his colleagues compared data on Hyperion to a theoretical criterion for chaos, the resonance overlap condition. This predicted that Hyperion’s orbit should interact chaotically with its spin, a prediction confirmed by solving the equations of motion numerically. The chaos in Hyperion’s dynamics manifests itself mainly as erratic tumbling. The orbit itself doesn’t vary as wildly. It’s like an American football rolling round and round an athletics track, sticking to one lane but tumbling unpredictably end over end. In 1984 the only known moon of Pluto was Charon, discovered in 1978, and no one could measure its rate of spin. The other four were discovered between 2005 and 2012. All five are crammed into an unusually small zone, and it’s thought that they were originally all part of a single larger body, which collided with Pluto during the early formation of the solar system – a miniature version of the giant impact theory of the formation of our own Moon. Charon is large, round, and tidally locked in a 1:1 resonance, so it always presents the same face to Pluto, just as the Moon does to the Earth. However, unlike the Earth, Pluto also always presents the same face to its moon. The tidal locking, and the round shape, prevent chaotic tumbling.

The other four moons are small, irregular, and are now known to tumble chaotically, like Hyperion. Plutonian numerology doesn’t stop with that 1:1 resonance. To a good approximation, Styx, Nix, Kerberos, and Hydra are in 1:3, 1:4, 1:5, and 1:6 orbital resonances with Charon; that is, their periods are roughly 3, 4, 5, and 6 times as long as Charon’s. However, those figures are only averages. The actual orbital periods vary significantly from one revolution to the next. Even so, in astronomical terms it all looks very orderly. Because order can give rise to chaos, it’s common for them both to coincide in the same system: orderly in some respects, chaotic in others. The two main research groups that work on chaos and the long-term dynamics of the solar system are headed by Wisdom and Laskar. In 1993, within a week of each other, both groups published papers describing a new cosmic context for chaos: the axial tilt of the planets. In Chapter 1 we saw that a rigid body spins about an axis: a line through the body that is instantaneously stationary. The spin axis can move over time, but in the short term it stays pretty much fixed. So the body spins like a top, with the axis as the central spindle. Planets, being almost spherical, spin at a very regular rate about an axis that seems not to change, even over centuries. In particular, the angle between the axis and the ecliptic plane, technically known as the obliquity, remains constant. For the Earth it’s 23·4 degrees. However, appearances are deceptive. Around 160 BC Hipparchus discovered an effect known as the precession of the equinoxes. In the Almagest, Ptolemy states that Hipparchus observed the positions in the night sky of the star Spica (alpha Virginis) and others. Two predecessors had done the same: Aristillus around 280 BC and Timocharis around 300 BC. Comparing data, Ptolemy concluded that Spica had drifted by about two degrees when observed at the autumnal equinox – the time at which night and day are equally long. He deduced that the equinoxes were moving along the zodiac at about one degree every century, and would eventually get back to where they started after 36,000 years.

We now know he was right, and why. Rotating bodies precess: their spin axis slowly changes direction, as the tip of the axis describes a slow circle. Spinning tops often do this. Mathematics going back to Lagrange explains precession as the typical dynamics of a body with a certain type of symmetry – two equal axes of inertia. Planets are approximately ellipsoids of rotation, so they satisfy this condition. The Earth’s axis precesses with a period of 25,772 years. This affects how we see the night sky. At the moment, the pole star, Polaris, in Ursa Major is aligned with the axis and it therefore appears to be fixed, while the rest of the stars seem to rotate around it. Actually, it’s the Earth that’s rotating. But in ancient Egypt, 5000 years ago, Polaris went round in a circle, and the faint star Batn al Thuban (phi Draconis) was fixed instead. I chose that date because it’s a matter of luck whether there’s a bright star near the pole or not, and mostly there isn’t. When a planet’s axis precesses, its obliquity doesn’t change. The seasons slowly drift, but so slowly that only a Hipparchus would notice, and only then with the help of previous generations. A given location on the planet experiences much the same seasonal variations, but their timing changes very slowly. Both Laskar’s and Wisdom’s groups discovered that Mars is different. Its obliquity also varies, to some extent driven by changes in its orbit. If the precession of its axis resonates with the period of any variable orbital element, the obliquity can change. The two groups calculated what effect this has by analysing the planet’s dynamics. Wisdom’s calculations show that the obliquity of Mars varies chaotically, ranging between 11 and 49 degrees. It can change by 20 degrees in about 100,000 years, and it oscillates chaotically over that sort of range at about that rate. Nine million years ago the obliquity varied between 30 and 47 degrees, and this continued until 4 million years ago, when there was a relatively abrupt shift to a range between 15 and 35 degrees. The calculations include effects from general relativity, which in this particular problem are important. Without those, the model doesn’t lead to this transition. The reason for the transition is – you’ve guessed it – passage through a spin–orbit resonance. Laskar’s group used a different model, without relativistic effects but with a more accurate representation of the dynamics, and examined a longer period of time. The group obtained similar results for Mars, but found that

over longer periods its obliquity varies between 0 and 60 degrees, an even wider range. They also studied Mercury, Venus, and Earth. Today, Mercury spins very slowly, once every 58 days, and it goes round the Sun in 88 days – a 3:2 spin–orbit resonance. This was probably caused by tidal interactions with the Sun, which slowed the primordial spin down. Laskar’s group calculated that orginally Mercury spun once every 19 hours. Before the planet reached its current state, its obliquity varied between 0 and 100 degrees, taking about a million years to cover most of that range. In particular, there were times when its pole faced the Sun. Venus poses a puzzle for astronomers, because, by the usual conventions about angles for spinning bodies, its obliquity is 177 degrees – essentially upside down. This causes it to rotate very slowly (period 243 days) in the opposite direction to every other planet. The explanation for this ‘retrograde’ motion isn’t known, but in the 1980s it was thought to be primordial: going right back to the origin of the solar system. Laskar’s analysis suggests this might not be the case. It’s thought that Venus originally had a rotation period of a mere 13 hours. Assuming this, the model shows that the obliquity of Venus originally varied chaotically, and when it reached 90 degrees it could have become stable rather than chaotic. From that state, it could gradually evolve to its present value. The results for the Earth are interestingly different. Earth’s obliquity is very stable, varying by only a degree. The reason seems to be our unusually large Moon. Without it, Earth’s obliquity would wander around between 0 and 85 degrees. On this alternative Earth, climatic conditions would be very different. Instead of the equator being warm and the poles cold, different regions would experience entirely different ranges of temperature. This would affect the weather patterns. Some scientists have suggested that without the Moon the chaotic changes in climate would have made it harder for life, especially complex life, to evolve here. However, life evolved in the oceans. It didn’t invade the land until about 500 million years ago. Marine life would not be greatly affected by a changing climate. As for land animals, the climatic changes that would result from the absence of the Moon are fast on astronomical timescales, but land organisms would migrate as the climate changed, because on their timescale the changes are slow. Evolution would proceed

largely unhindered. It might even be speeded up by stronger pressure to adapt. Astronomical effects on Earth’s living creatures that actually happened are more interesting than hypothetical ones that didn’t. The most famous is the asteroid that destroyed the dinosaurs. Or was it a comet? And were other influences involved too, such as massive volcanic eruptions? Dinosaurs first appeared about 231 million years ago in the Triassic, and disappeared 65 million years ago at the end of the Cretaceous. In between, they were the most successful vertebrates, in sea and on land. By comparison, ‘modern’ humans have been around for about 2 million years. However, there were many species of dinosaur, so that’s a bit unfair. Most individual species survive for no more than a few million years. The fossil record shows that the dinosaurs died out very suddenly by geological standards. Their demise was accompanied by that of mosasaurs, plesiosaurs, ammonites, many birds, most marsupials, half the types of plankton, many fishes, sea urchins, sponges, and snails. This ‘K/T extinction’ is one of five or six major events in which huge numbers of species perished in a geological eye-blink.12 The dinosaurs did manage to leave some modern descendants, though: birds evolved from theropod dinosaurs in the Jurassic. Towards the end of their reign dinosaurs coexisted with mammals, some quite large, and the disappearance of dinosaurs seems to have triggered a burst of mammalian evolution as the main competition was removed from the scene. There’s a widespread consensus among palaeontologists that a major cause of the K/T extinction was the impact of an asteroid, or possibly a comet, which left an indelible mark on the Yucatan coast of Mexico: the Chicxulub crater. Whether this was the sole cause is still contentious, partly because there’s at least one other plausible candidate: the massive volcanic outpourings of magma that formed the Deccan Traps in India, which would have sent large amounts of noxious gases into the atmosphere. ‘Traps’ here comes from the Swedish for ‘stairs’ – the basalt strata tend to weather into a series of steps. Maybe climate change or changing sea levels were involved

too. But the impact is still the prime suspect, and several attempts to prove otherwise have foundered as improved evidence came in. The main problem with the Deccan Traps theory, for instance, is that they formed over a period of 800,000 years. The K/T extinction was much more rapid. In 2013 Paul Renne used argon–argon dating (a comparison of proportions of different isotopes of the gas argon) to pin down the impact to 66·043 million years ago, plus or minus 11,000 years. The death of the dinosaurs seems to have happened within 33,000 years of that date. If correct, the timing seems too close to be a coincidence. But it’s certainly possible that other causes stressed the world’s ecosystem, and the impact was the coup de grâce. In fact, in 2015 a team of geophysicists led by Mark Richards found clear evidence that shortly after the impact, the flow of lava from the Deccan traps doubled.13 This adds weight to an older theory: the impact sent shockwaves round the Earth. They focused on the region diametrically opposite Chicxulub, which happens to be very close to the Deccan traps. Astronomers have tried to find out whether the impactor was a comet or an asteroid, and even where it came from. In 2007 William Bottke and others14 published an analysis of chemical similarities suggesting that the impactor originated in a group of asteroids known as the Baptinista family, and that this broke up about 160 million years ago. But at least one asteroid from this group has the wrong chemistry, and in 2011 the timing of the break-up was estimated as 80 million years, which doesn’t leave a long enough gap before the impact. One thing that has been established is how chaos causes asteroids to be flung out of their belt and end up hitting the Earth. The culprit is Jupiter, ably assisted by Mars. Recall from Chapter 5 that the asteroid belt has gaps – distances from the Sun where the population is unusually sparse – and that these correlate well with orbits in resonance with Jupiter. In 1983 Wisdom studied the formation of the 3:1 Kirkwood gap, seeking to understand the mathematical mechanism that causes asteroids to be ejected from such an orbit. Mathematicians and physicists had already discovered a close association

between resonance and chaos. At the heart of a resonance is a periodic orbit, in which the asteroid makes a whole number of revolutions while Jupiter makes another whole number. Those two numbers characterise the resonance, and in the above example they are 3 and 1. However, such an orbit will change because other bodies perturb the asteroid. The question is: how? In the mid twentieth century three mathematicians – Andrei Kolmogorov, Vladimir Arnold, and Jürgen Moser – obtained different bits of the answer to this question, collected together in the KAM theorem. This states that orbits near the periodic one are of two kinds. Some are quasiperiodic, spiralling around the original orbit in a regular manner. The others are chaotic. Moreover, the two types are nested in an intricate manner. The quasiperiodic orbits spiral around tubes that surround the periodic orbit. There are infinitely many of these tubes. Between them are more complicated tubes, spiralling around the spiral orbits. Between those are even more complicated tubes spiralling around those, and so on. (This is what ‘quasiperiodic’ means.) The chaotic orbits fill the intricate gaps between all of these spirals and multiple spirals, and are defined by Poincaré’s homoclinic tangles. This highly complex structure can most easily be visualised by borrowing a trick from Poincaré and looking at them in cross section. The initial periodic orbit corresponds to the central point, the quasiperiodic tubes have the closed curves as cross sections, and the shaded regions between them are traces of chaotic orbits. Such an orbit passes through some point in the shaded region, travels all the way round near the original periodic orbit, and hits the cross section again at a second point – whose relation to the first appears random. What you’d observe wouldn’t be an asteroid performing a drunkard’s walk; it would be an asteroid whose orbital elements change chaotically from one orbit to the next.

Numerically computed cross section of orbits near a periodic one, in accordance with the KAM theorem. To carry out specific computations for the 3:1 Kirkwood gap, Wisdom invented a new method to model the dynamics: a formula that matches how successive orbits hit the cross section. Instead of solving a differential equation for the orbit, you just keep applying the formula. The results confirm that chaotic orbits occur, and provide details of what they look like. For the most interesting ones, the eccentricity of the approximate ellipse suddenly grows much larger. So an orbit that’s reasonably close to a circle, maybe a fattish ellipse, turns into a long thin one. Long enough, in fact, to cross the orbit of Mars. Since it keeps doing that, there’s good chance that it will come close to Mars, and be perturbed by the slingshot effect. And that would fling it … anywhere. Wisdom suggested that this mechanism is how Jupiter clears out the 3:1 Kirkwood gap. As confirmation, he plotted the orbital elements of asteroids near the gap, and compared them to the chaotic zone of his model. The fit is near perfect. Basically, an asteroid trying to orbit in the gap gets shaken up by the chaos and gets passed to Mars, which kicks it away. Jupiter takes a corner kick, Mars scores. And sometimes … just sometimes … Mars kicks it in our direction. And if the kick happens to be on target –

Spike in eccentricity (vertical axis). Horizontal axis is time. Outer edges of the chaotic zone (solid lines) and orbital elements of asteroids (dots and crosses). Vertical axis is eccentricity, horizontal axis is major radius relative to that of Jupiter. Mars one, dinosaurs nil.

10 The Interplanetary Superhighway Space travel is utter bilge. Richard Woolley, Astronomer Royal, 1956 WHEN VISIONARY SCIENTISTS and engineers first started thinking seriously about landing humans on the Moon, one of the first problems was to work out the best route. ‘Best’ has many meanings. In this instance the requirements are a fast trajectory, minimising the time vulnerable astronauts spend hurtling through vacuum in a glorified tin can, and switching the rocket engine on and off as few times as possible to reduce the chance of it failing. When Apollo 11 landed two astronauts on the Moon, its trajectory obeyed these two principles. First, the spacecraft was injected into low Earth orbit, where everything could be checked to make sure it was still functional. Then a single burst of the engines sent it speeding towards the Moon. When it got close, a few more bursts slowed it down again, injecting it into lunar orbit. The landing module then went down to the surface, and its top half came back a few days later with the crew. It was then jettisoned, and the crew returned to Earth with another burst from the engine to take them out of lunar orbit. After coasting home they came to the most dangerous part of the entire mission: using friction with the Earth’s atmosphere as a brake, to slow the command capsule down enough for it to land using parachutes. For a time, this type of trajectory, which in its simplest form is known as a Hohmann ellipse, was used for most missions. There’s a sense in which the Hohmann ellipse is optimal. Namely, it’s faster than most alternatives, for the same amount of rocket fuel. But as humanity gained experience with space missions, engineers realised that other types of mission have different requirements. In particular, speed is less important if you’re sending a machine or supplies.

Hohmann ellipse. Thick line shows transfer orbit. Until 1961 mission planners, convinced that a Hohmann ellipse is optimal, viewed the gravitational field of a planet as an obstacle, to be overcome using extra thrust. Then Michael Minovitch discovered the slingshot effect in a simulation.1 Within a few decades, new ideas from the mathematics of many-body orbits led to the discovery that a spacecraft can get to its destination using far less fuel, by following a trajectory very different from that used for the Moon landing. The price is that it takes much longer, and may require a more complex series of rocket boosts. However, today’s rocket engines are more reliable, and can be fired repeatedly without greatly increasing the likelihood of failure. Instead of considering just the Earth and the final target, engineers started thinking about all of the bodies that might potentially affect a space probe’s trajectory. Their gravitational fields combine to create a kind of energy landscape, a metaphor that we encountered in connection with Lagrange points and Greek and Trojan asteroids. The spacecraft in effect wanders around the contours of this landscape. One twist is that the landscape changes as the bodies move. Another is that mathematically this is a landscape in many dimensions, not just the usual three, because velocity is important as well as position. A third is that chaos plays a key role: you can take advantage of the butterfly effect to obtain large results from small causes. These ideas have been used in real missions. They also imply that the solar system has a network of invisible mathematical tubes linking its planets, an interplanetary superhighway system providing unusually efficient routes between them.2 The dynamics governing these tubes may

even explain how the planets are spaced, a modern advance on the Titius– Bode law. Artist’s conception of the interplanetary superhighway. The ribbon represents one possible trajectory along a tube, and constrictions represent Lagrange points. The Rosetta mission is an example of new ways to design trajectories for space probes. It doesn’t use the butterfly effect, but it shows how imaginative planning can produce results that at first seem impossible, by exploiting natural features of the solar system’s gravitational landscape. Rosetta was technically challenging, not least because of the distance and speed of the target. At the time of the landing, comet 67P was 480 million kilometres from Earth and travelling at over 50,000 kph. That’s sixty times as fast as a passenger jet aircraft. Because of the limitations of current rocketry, the point-and-go method used for the Moon landing won’t work. Getting out of Earth orbit with enough speed is difficult and expensive, but it’s possible. Indeed, the New Horizons mission to Pluto took the direct

route. It did borrow some extra velocity from Jupiter along the way, but it could have got there without that by taking longer. The big problem was slowing down again; this was solved by not even trying. New Horizons, the fastest space vehicle ever launched, used a very powerful rocket with five solid fuel boosters and an extra final stage to get up to speed when leaving the Earth. It also left them behind as soon as it could: too heavy to keep, and empty of fuel anyway. When the probe got to Pluto it barrelled through the system at high speed, and had to do all of its main scientific observations within a period of about a day. During that time it was too busy to communicate with Earth, causing a nervous period while the mission scientists and controllers waited to see whether it had survived the encounter – colliding with a single dust grain could have proved fatal. In contrast, Rosetta had to rendezvous with 67P and stay with it as the comet neared the Sun, observing it all the time. It had to deposit Philae on the comet’s surface. Relative to the comet, Rosetta had to be pretty much stationary, but the comet was 300 million miles away and moving at a colossal speed – 55,000 kph. So the mission trajectory had to be designed to bring it up to speed, yet end up in the same orbit as the comet. Even finding a suitable trajectory was difficult; so was finding a suitable comet. In the event, the probe followed a highly indirect route,3 which among other things returned near the Earth three times. It was a bit like travelling from London to New York by first shuttling back and forth several times between London and Moscow. But cities stay still relative to the Earth, whereas planets don’t, and that makes all the difference. The probe began its epic journey by moving in what naively appears to be totally the wrong direction. It headed towards the Sun, even though the comet was far outside the orbit of Mars, and moving away. (I don’t mean directly towards: just that the distance to the Sun was becoming shorter.) Rosetta’s orbit swung past the Sun and returned close to the Earth, where it was flung outwards to an encounter with Mars. It then swung back to meet the Earth for a second time, then back out beyond Mars’s orbit again. By now the comet was on the far side of the Sun and closer to it than Rosetta was. A third encounter with Earth flung the probe outwards again, chasing the comet as it now sped away from the Sun. Finally, Rosetta made its rendezvous with destiny. Why such a complicated route? ESA didn’t just point its rocket at the comet and blast off. That would have required far too much fuel, and by the

time it got there, the comet would have been somewhere else. Instead, Rosetta performed a carefully choreographed cosmic dance, tugged by the combined gravitational forces of the Sun, the Earth, Mars, and other relevant bodies. Its route, calculated by exploiting Newton’s law of gravity, was designed for fuel efficiency. Each close fly-by with Earth and Mars gave the probe a free boost as it borrowed energy from the planet. An occasional small burst from four thrusters kept the craft on track. The price paid for conserving fuel was that Rosetta took ten years to get to its destination. However, without paying that price, the mission would have been far too costly to get off the ground at all. This kind of trajectory, going round and round and in and out, seeking judicious speed boosts from planets and moons, has become commonplace for space missions when time is not of the essence. If a space probe passes close behind a planet as it travels along its orbit, the probe can steal some of the planet’s energy in a slingshot manoeuvre. The planet actually slows down, but the decrease is too small for even the most sensitive apparatus to observe. So the probe gets a boost in speed without having to use up any rocket fuel. The devil, as always, is in the detail. In order to design such trajectories, the engineers must be able to predict the movements of all bodies involved, and they have to make the whole journey fit together to get the probe to its intended destination. So mission design is a mixture of calculation and black art. Everything depends on an area of human activity whose role in space exploration is seldom even hinted at, but without which, nothing could be achieved. Whenever the media start talking about ‘computer models’ or ‘algorithms’, you can presume that they really mean ‘mathematics’, but are either too scared to mention the word, or think it will scare you. There are sensible reasons not to rub people’s noses in complex mathematical detail, but it does a grave disservice to one of humanity’s most powerful ways of thinking to pretend it’s not there at all. Rosetta’s main dynamic trick was the slingshot manoeuvre. Aside from those repeated encounters, it effectively followed a series of Hohmann ellipses. Instead of going into orbit around 67P, it followed a nearby ellipse